In this paper we give an explicit parametrization for all two-qubit density matrices. This is important for calculations involving entanglement and many other types of quantum information processing. To accomplish this we present a generalized Euler angle parametrization for SU(4) and all possible two-qubit density matrices. The important group-theoretical properties of such a description are then manifest. We thus obtain the correct Haar (Hurwitz) measure and volume element for SU(4) which follows from this parametrization. In addition, we study the role of this parametrization in the Peres–Horodecki criteria for separability and its corresponding usefulness in calculating entangled two-qubit states as represented through the parametrization. PACS numbers: 03.67.−a, 02.20.−a, 03.65.Ud 1.
In quantum mechanics, sets of density matrices are important for numerous reasons. For example, their compact notation make them useful for describing decoherence and entanglement properties of multi-particle quantum systems. In particular, two two-state density matrices, otherwise known as two qubit density matrices, are important for their role in explaining quantum teleportation, dense coding, and computation theorems. The aim of this paper is to show an explicit parameterization for the Hilbert space of all two qubit density matrices. Such a parameterization would be extremely useful for numerical calculations concerning entanglement and other quantum information parameters. We would also like to know the properties of such parameterized two qubit density matrices; in particular, the representation of their convex sets, their sub-sets, and their set boundaries in terms of our parameterization. Here we present a generalized Euler angle parameterization for SU(4) and all possible two qubit density matrices as well as the corrected Haar Measures for SU(3) and SU(4) from this parameterization. The role of the parameterization in the Peres-Horodecki criteria will also be introduced as well as its usefulness in calculating entangled two qubit states.
In a previous paper [1] an Euler angle parametrization for SU (4) was given. Here we present the derivation of a generalized Euler angle parametrization for SU ( N ). The formula for the calculation of the Haar measure for SU ( N ) as well as its relation to Marinov's volume formula for SU ( N ) [2, 3] will also be derived. As an example of this parametrization's usefulness, the density matrix parametrization and invariant volume element for a qubit/qutrit, three qubit and two three-state systems, also known as two qutrit systems [4], will also be given.
Entanglement concurrence is an important bipartite entanglement measure that has found wide applications in quantum technologies. In this work, inspired by unified entropy, we introduce a two-parameter family of entanglement measures, referred to as the unified
(q,s)-concurrence. Both the standard entanglement concurrence and the recently proposed
q-concurrence emerge as special cases within this family. By combining the positive partial transposition and realignment criteria, we derive an analytical lower bound for this measure for arbitrary bipartite mixed states, revealing a connection to strong separability criteria. Explicit expressions are obtained for the unified
(q,s)-concurrence in the cases of isotropic and Werner states under the constraint
q>1 and
qs≥1. Furthermore, we explore the monogamy properties of the unified
(q,s)-concurrence for
q≥2,
0≤s≤1 and
1≤qs≤3, in qubit systems. In addition, we derive an entanglement polygon inequality for the unified
(q,s)-concurrence with
q≥1 and
qs≥1, which manifests the relationship among all the marginal entanglements in any multipartite qudit system.
A new geometric representation of qubit and qutrit states based on probability simplexes is used to describe the separability and entanglement properties of density matrices of two qubits. The Peres-Horodecki positive partial transpose (ppt) -criterion and the concurrence inequalities are formulated as the conditions that the introduced probability distributions must satisfy to present entanglement. A four-level system, where one or two states are inaccessible, is considered as an example of applying the elaborated probability approach in an explicit form. The areas of three Triadas of Malevich's squares for entangled states of two qubits are defined through the qutrit state, and the critical values of the sum of their areas are calculated. We always find an interval for the sum of the square areas, which provides the possibility for an experimental checkup of the entanglement of the system in terms of the probabilities.